# Katzs middle convolution algorithm

1 JAD - Laboratoire Jean Alexandre Dieudonné

Abstract : This is an expository account of Katz-s middle convolution operation on local systems over ${\bf P}^1-\{ q 1,\ldots , q n\}$. We describe the Betti and de Rham versions, and point out that they give isomorphisms between different moduli spaces of local systems, following Völklein, Dettweiler-Reiter, Haraoka-Yokoyama. Kostov-s program for applying the Katz algorithm is to say that in the range where middle convolution no longer reduces the rank, one should give a direct construction of local systems. This has been done by Kostov and Crawley-Boevey. We describe here an alternative construction using the notion of cyclotomic harmonic bundles: these are like variations of Hodge structure except that the Hodge decomposition can go around in a circle.

Author: Carlos Simpson -

Source: https://hal.archives-ouvertes.fr/